In the first part of this article, Feferman outlines his ‘conceptual structuralism’ and emphasizes broad similarities between Parsons’s and his own structuralist perspective on mathematics. However, Feferman also notices differences and makes two critical claims about any structuralism that focuses on the “ur-structures” of natural and real numbers: (1) it does not account for the manifold use of other important structures in modern mathematics and, correspondingly, (2) it does not explain the ubiquity of “individual [natural or real] numbers” in that use. In the second part, Feferman presents a summary of his reasons for the skepticism he has towards contemporary metamathematical investigations of set theory. That skepticism led him to reject the Continuum Problem as a definite mathematical one. He contrasts that attitude sharply to Parsons’s “great sympathy for the current explorations of higher set theory.”

In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear'” and “completely definite,”' many of the statements of analysis and set theory are “inherently vague'” and “indefinite.”' I critique his four central arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to this final, remaining point will be that the concept of “being clear enough to secure definiteness” is about as clear a case of an inherently vague and indefinite concept as one might find, and as such it can bear little weight in making a case against the definiteness of analysis and set theory.

An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism” (Tait 1981), namely the idea of constructions f(X) from ‘arbitrary’ or ‘generic’ objects of various types (triangles, natural numbers, etc.). The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number (as a common noun) and arithmetic, he was not referring to the natural or whole numbers and their arithmetic, but rather to the real numbers (as then understood) and their arithmetic. (This thesis owes, and will receive, some account of Kant’s discussion of number as the schema of magnitude.)

In his “Kant and Finitism” Tait attempts to connect his analysis of finitist arithmetic with Kant’s perspective on arithmetic. The examination of this attempt is the basis for a distinctive view on the dramatic methodological shift from Kant to Dedekind and Hilbert. Dedekind’s 1888 essay “Was sind und was sollen die Zahlen?” gives a logical analysis of arithmetic, whereas Hilbert’s 1899 book “Grundlagen der Geometrie” presents such an analysis of geometry or, as Hilbert puts it, of our spatial intuition. This shift in the late ninteenth century required a radical expansion of logic: first by the inclusion of principles for “systems” (sets) and “mappings” (functions), but second by a structuralist broadening of axioms and inferential principles. The interaction of mathematics and logic in mathematical logic opened, around 1920, fields of investigation with enormous impact on the philosophy of mathematics, promoting a deeper integration of mathematical practice and philosophical reflection.

I comment on Feferman’s views on set theory, in particular criticizing a priori arguments claiming that the continuum hypothesis has no determinate truth value and commenting on his responses to my paper on his skepticism about set theory. I respond to criticisms of his of the structuralism that I have advocated and comment on his view of proof theory. On Koellner’s paper, I register little disagreement but note a difference of sympathy about views such as constructivism. On Tait’s paper, I note that Kant gives more play to the notion of whole number than Tait seems to allow and that Kant’s conception of real numbers is unclear. Responding to Sieg’s paper, I note his emphasis on how much mathematics and its foundations changed from Kant’s time to that of Dedekind and Hilbert and mention my effort to find a limited role for an intuition distantly descended from Kant’s.